Bounds for the minimal solution of genus zero diophantine equations

LXXXVI.1 (1998)

Bounds for the minimal solution of genus zero diophantine equations

by

Dimitrios Poulakis (Thessaloniki)

1. Introduction. In [8], Holzer proved that if the equation aX

+ bY

+ cZ

= 0, where a, b, c ∈ Z, has a non-trivial solution in integers, then a so- lution (x, y, z) exists with |x| ≤ |bc|

, |y| ≤ |ac|

, |z| ≤ |ab|

. Later, Mordell [12] gave a simple elementary proof of this result. Let K be an algebraic number field. In case where a, b, c are integers of K, Siegel [20]

obtained a very sharp estimate for the size of the “smallest” solution of the above equation in integers of K. In this work we generalize these results.

Let F (X, Y ) be an absolutely irreducible polynomial of K[X, Y ] such that the equation F (X, Y ) = 0 defines a curve C of genus 0. Suppose that C has a non-singular point defined over K. Then we calculate an explicit up- per bound for the size of the “smallest” non-singular point of C over K.

Furthermore, we obtain an effective parametrization of C.

A fundamental result due to Hilbert and Hurwitz [6] says that any curve of genus 0 defined over Q is birationally equivalent to either a line or a conic.

The same result was obtained independently by Poincar´e [13]. Furthermore, in [13], Poincar´e proved, by another method, that any curve of genus 0 defined over Q is birationally equivalent to a conic. In Sections 3 and 4 we give an effective proof of these results. In Section 3, we deal with curves of genus 0 defined over K with only ordinary singular points. We prove that every curve of this class is birationally equivalent over K to a conic, giving explicit estimates on the size of the conic, the birational isomorphism and its inverse. In the case where the curve has odd degree we prove that it is birationally equivalent over K to a line giving explicit estimates for the birational isomorphism and its inverse.

A classical result asserts that any curve A is birationally equivalent to a plane curve E with at most ordinary double points as singularities. In Section 4, we give an effective proof of this result for the case of curves of

1991 Mathematics Subject Classification: 11G30, 14H25, 11D41.

[51]

genus 0 defined over K and we obtain explicit estimates about the size of E, the birational isomorphism and its inverse. Finally, in Section 5, Siegel’s estimate for the size of the “smallest” solution of equation aX

+bY

+cZ

= 0 and the results of Sections 3 and 4 imply an upper bound for the size of the

“smallest” solution over K of equations defining curves of genus 0 over K.

Moreover, these results give an effective parametrization of curves of genus 0.

Hence, if we know that a curve of genus 0 defined over K has a non-singular point over K, then we have an effective characterization of all its points over K.

2. Statement of the main results. Let K be an algebraic number field of degree d and of discriminant D

. We consider the set of standard absolute values on Q containing the ordinary absolute value |·| and for every prime p the p-adic absolute value | · |

. If x = p

a/b, where a, b are integers not divisible by p, then by definition |x|

= p

. By an absolute value of K we will always understand an absolute value that extends one of the above absolute values of Q. We denote by M (K) a set of symbols v such that with every v ∈ M (K) there is associated precisely one absolute value | · |

on K.

For every v ∈ M (K) we denote by K

the completion of K at v and by d

the degree of K

over Q

. Let x = (x

: . . . : x

) be a point of the projective space P

(K) over K. We define the field height H

(x) of x by

H

(x) = Y

v∈M (K)

max{|x

|

, . . . , |x

|

}

,

and the absolute height H(x) by H(x) = H

(x)

. Further, for x ∈ K we define H

(x) = H

((1 : x)) and H(x) = H((1 : x)). Let G be a polynomial in one or several variables and with coefficients in K. We define the field height H

(G) and the absolute height H(G) of G to be respectively the field height and the absolute height of the point in a projective space having as coordinates the coefficients of G (in any order). Given v ∈ M (K), we denote by |G|

the maximum of |c|

over all the coefficients c of G. For an account of the properties of heights see [21, Chap. VIII; 10, Chap. 3].

Let us now state our main results.

Theorem 2.1. Let F (X, Y ) be an absolutely irreducible polynomial in K[X, Y ] of degree N ≥ 3 such that the curve C defined by the equation F (X, Y ) = 0 is of genus 0. Then there is a conic Γ defined over K of equation G(X, Y ) = 0 with

H(G) < (9N

H(F ))

and a birational map Φ : C → Γ given by

Φ(X, Y ) =

 φ

(X, Y )

φ

(X, Y ) , φ

(X, Y ) φ

(X, Y )



,

where φ

(X, Y ) ∈ K[X, Y ] (i = 1, 2, 3) with deg φ

< 3N

and H(φ

) < (9N

H(F ))

(i = 1, 2, 3).

The inverse map of Φ is given by Φ

(X, Y ) =

 τ

(X, Y )

τ

(X, Y ) , τ

(X, Y ) τ

(X, Y )

 ,

where τ

(X, Y ) ∈ K[X, Y ] (i = 1, 2, 3, 4) with deg τ

< 15N

and H(τ

) <

 (9N

H(F ))

(i = 1, 3), (N + 1)

H(F )

(i = 2, 4).

Theorem 2.2. Let F (X, Y ) be an absolutely irreducible polynomial in K[X, Y ] of odd degree N ≥ 3 such that the curve C defined by the equation F (X, Y ) = 0 is of genus 0. Then there is a birational map Ψ : C → P

given by

Ψ (X, Y ) = ψ

(X, Y ) ψ

(X, Y ) ,

where ψ

(X, Y ) ∈ K[X, Y ] (i = 1, 2) with deg ψ

< 3N

and H(ψ

) < (9N

H(F ))

. The inverse map of Ψ is given by

X = σ

(T )

σ

(T ) , Y = σ

(T ) σ

(T ) , where σ

(T ) ∈ K[T ] (i = 1, 2, 3, 4) with deg σ

< 8N

and

H(σ

) <

 (9N

H(F ))

(i = 1, 3), (N + 1)

H(F )

(i = 2, 4).

Theorem 2.3. Let F (X, Y, Z) be a homogeneous absolutely irreducible polynomial in K[X, Y, Z] of degree N ≥ 2 such that the curve C defined by the equation F (X, Y, Z) = 0 is of genus 0. Suppose that C has a non-singular point defined over K. Then there exists a non-singular point P of C defined over K such that

H(P ) < |D

|

(9N

H(F ))

. Moreover , the curve F (X, Y, 1) = 0 has a parametrization given by

X = g

(T )

g

(T ) , Y = g

(T ) g

(T ) , where g

(T ) ∈ K[T ] (i = 1, 2, 3, 4) with deg g

< 30N

and

H(g

) < |D

|

(9N

H(F ))

.

3. Curves with only ordinary singular points

3.1. Statement of the results. In this section we give an effective proof of the fact that a curve with only ordinary singular points is birationally equivalent to a conic. Our method develops some arguments that go back to some ideas of Poincar´e [13]. Furthermore, a variant of our method gives an effective proof of the fact that a curve with only ordinary singular points and odd degree is birationally equivalent to a line. More precisely, we prove the following results:

Theorem 3.1. Let F (X, Y ) be an absolutely irreducible polynomial in K[X, Y ] of degree N ≥ 3 such that the curve C defined by the equation F (X, Y ) = 0 is of genus 0. Suppose that C has only ordinary multiple points.

Then there is a conic Γ defined over K of equation G(X, Y ) = 0 with H(G) < (N + 1)

H(F )

and a birational map Ψ : C → Γ defined by Ψ (X, Y ) =

 ψ

(X, Y )

ψ

(X, Y ) , ψ

(X, Y ) ψ

(X, Y )

 ,

where ψ

(X, Y ) ∈ K[X, Y ] (i = 1, 2, 3) with deg ψ

≤ N − 1 and H(ψ

) < (N + 1)

H(F )

(i = 1, 2, 3).

The inverse map of Ψ is given by Ψ

(X, Y ) =

 ω

(X, Y )

ω

(X, Y ) , ω

(X, Y ) ω

(X, Y )

 ,

where ω

(X, Y ) ∈ K[X, Y ] (i = 1, 2, 3, 4) with deg

ω

≤ N , deg

ω

≤ N and

H(ω

) < (N + 1)

H(F )

.

Theorem 3.2. Let F (X, Y ) be an absolutely irreducible polynomial in K[X, Y ] of odd degree N ≥ 3 such that the curve C defined by the equation F (X, Y ) = 0 is of genus 0. Suppose that C has only ordinary multiple points.

Then there is a birational map Ψ : C → P

defined by Ψ (X, Y ) = ψ

(X, Y )

ψ

(X, Y ) ,

where ψ

(X, Y ) ∈ K[X, Y ] (i = 1, 2) with deg ψ

≤ N − 2 and H(ψ

) < (N + 1)

H(F )

(i = 1, 2).

The inverse map of Ψ is given by X = ω

(T )

ω

(T ) , Y = ω

(T )

ω

(T ) ,

where ω

(T ) ∈ K[T ] (i = 1, 2, 3, 4) with deg ω

≤ N and

H(ω

) < (N + 1)

H(F )

(i = 1, 2, 3, 4).

3.2. Auxiliary lemmas. We give some lemmas which will be useful for the proof of our results. We prove only those which are not yet in the literature.

Lemma 3.1. Let F (X) = c

X

+ c

X

+ . . . + c

be a polynomial in K[X] − K and let α be one of its roots. Then

H(α) < 2H(F ).

P r o o f. See [11; 14, Lemma 4].

Lemma 3.2. Let P (X, Y, V ), Q(X, Y, W ) ∈ K[X, Y, V, W ] − K. Denote by R(X, V, W ) the resultant of P (X, Y, V ) and Q(X, Y, W ), considered as polynomials with coefficients in K[X, V, W ]. Put deg

P = m

, deg

P = n

, deg

P = r

and deg

Q = m

, deg

Q = n

, deg

Q = r

. Assume R(X, V, W ) 6= 0. Then

H(R) ≤ (n

+ n

)!((r

+ 1)(m

+ 1))

((r

+ 1)(m

+ 1))

H(P )

H(Q)

. P r o o f. Write

P (X, Y, V ) = P

(X, V )Y

+ . . . + P

(X, V ), Q(X, Y, W ) = Q

(X, W )Y

+ . . . + Q

(X, W ),

where P

(X, V ) ∈ K[X, V ] (i = 0, . . . , n

) and Q

(X, W ) ∈ K[X, W ] (i = 0, . . . , n

). The polynomial R(X, V, W ) is homogeneous of degree n

in P

(X, V ), . . . , P

(X, V ) and of degree n

in Q

(X, W ), . . . , Q

(X, W ) with coefficients in Z. If | · |

is a non-archimedean absolute value, then

|R|

≤ |P |

|Q|

.

Let | · |

be an archimedean absolute value. If M (X, V, W ) is a monomial of degree n

in P

(X, V ), . . . , P

(X, V ) and of degree n

in Q

(X, W ), . . . . . . , Q

(X, W ), then

|M (X, V, W )|

≤ ((r

+ 1)(m

+ 1))

((r

+ 1)(m

+ 1))

|P |

|Q|

. Thus

|R|

≤ (n

+ n

)!((r

+ 1)(m

+ 1))

((r

+ 1)(m

+ 1))

|P |

|Q|

. Therefore

H(R) ≤ (n

+ n

)!((r

+ 1)(m

+ 1))

((r

+ 1)(m

+ 1))

H(P )

H(Q)

. Lemma 3.3. Let f and g be two polynomials of K[X

, . . . , X

] − K such that g(X) divides f (X). Then

H(g) ≤ 4

H(f ).

P r o o f. Let h be a polynomial in K[X

, . . . , X

] such that gh = f . By [10, Proposition 2.4, p. 57], we get

H(g)H(h) ≤ 4

H(f ).

The lemma follows.

If G(X, Y, Z) ∈ K[X, Y, Z], then by G

(X, Y, Z) we denote, as usual, the (a, b, c)-partial derivative of G(X, Y, Z) with respect to X, Y and Z.

Lemma 3.4. Let F (X, Y, Z) be an irreducible homogeneous polynomial in K[X, Y, Z]. Let P be a singular point of the projective curve F (X, Y, Z) = 0.

Then

H(P ) < 4(N + 1)

H(F )

. P r o o f. Suppose P = (a : b : 1). Then

F (a, b, 1) = F

(a, b, 1) = F

(a, b, 1) = 0.

We denote by R

(X) the resultant of F (X, Y, 1) and F

(X, Y, 1) with respect to Y and by R

(Y ) the resultant of F (X, Y, 1) and F

(X, Y, 1) with respect to X. Thus R

(a) = R

(b) = 0. Lemma 3.1 yields

H(P ) ≤ H(a)H(b) < 4H(R

)H(R

).

By Lemma 3.2,

H(R

) < N

(N + 1)

H(F )

(i = 1, 2).

Hence

H(P ) < 4(N + 1)

H(F )

.

Finally, if P = (a : b : 0), then H(P ) < 2H(F ). The lemma follows.

In the above proof we have used the inequality m! < ((m + 1)/2)

, for every positive integer m (see A. Cauchy, Exercices d’Analyse, Vol. 4, Paris, 1847, p. 106). Throughout the paper we shall use this inequality without further mention.

Lemma 3.5. Let F (X, Y ) be a polynomial in K[X, Y ] of degree m > 0 in X and n > 0 in Y . Let x, y ∈ K satisfy F (x, y) = 0 and deg F (x, Y ) = n.

Then

H(y) < 2(m + 1)H(F )H(x)

. P r o o f. See [15, Lemma 7].

Lemma 3.6. Let A

= (a

, . . . , a

) (i = 1, . . . , ν) be ν linearly inde-

pendent vectors in K

(ν < µ) and V be the K-vector space generated by

A

(i = 1, . . . , ν). Let G be the Galois group of K over K. Suppose that

σ(V ) = V for every σ ∈ G. Then there are µ − ν linearly independent vectors x

= (x

, . . . , x

) (i = 1, . . . , µ − ν) in K such that

H(x

) ≤ ν!H(A

) . . . H(A

) (i = 1, . . . , µ − ν), satisfying the linear system

a

X

+ . . . + a

X

= 0 (i = 1, . . . , ν).

P r o o f. Let A be the matrix with rows A

(i = 1, . . . , ν). We may sup- pose, without loss of generality, that the ν × ν-matrix ∆ formed by the ν first columns of A has rank ν. Thus |∆| 6= 0. We denote by ∆

(j = 1, . . . , ν, k = ν + 1, . . . , µ) the matrix obtained from ∆ by replacing the jth column by the kth column of A. Now the linear system is equivalent to

|∆|X

= −|∆

|X

− . . . − |∆

|X

(j = 1, . . . , ν).

Taking (X

, . . . , X

) = (−1, . . . , 0), . . . , (0, . . . , −1), we have respectively the solutions

x

=

 |∆

|

|∆| , . . . , |∆

|

|∆| , −1, 0, . . . , 0

 , .. .

x

=

 |∆

|

|∆| , . . . , |∆

|

|∆| , 0, . . . , 0, −1



which are linearly independent elements of K

.

Let σ ∈ G. Since σ(V ) = V , the vectors σ(A

) = (σ(a

), . . . , σ(a

)) (i = 1, . . . , ν) form a basis of V . Then there is an invertible ν × ν-matrix B such that

(σ(A

), . . . , σ(A

)) = (A

, . . . , A

)B.

If σ(∆) and σ(∆

) are the matrices obtained by the action of σ on the entries of ∆ and ∆

respectively, then σ(∆) = B

∆ and σ(∆

) = B

∆

(where B

is the transpose of B). It follows that

σ

 |∆

|

|∆|



= |σ(∆

)|

|σ(∆)| = |B| · |∆

|

|B| · |∆| = |∆

|

|∆| .

Hence, |∆

|/|∆| ∈ K (j = 1, . . . , ν, k = ν + 1, . . . , µ), whence x

∈ K

(i = 1, . . . , µ − ν).

The v-adic absolute value of a minor of A of order ν is

≤ |A

|

. . . |A

|

v(ν!),

where v(ν!) = ν! if | · |

is archimedean and v(ν!) = 1 otherwise. Thus,

H(x

) ≤ ν!H(A

) . . . H(A

) (i = 1, . . . , µ − ν).

Lemma 3.7. Let φ : C

→ C

be a rational map of algebraic curves.

Suppose that φ is defined and injective on an open subset U of C

. Then φ is a birational map.

P r o o f. Let e C

be a non-singular model of C

and f

be a birational morphism from e C

onto C. Then f

◦ φ ◦ f

: e C

→ e C

is a non-constant morphism of smooth curves and its restriction to the open set f

(U ) is injective. By [21, Proposition 2.6(b), p. 28], for all but finitely many Q ∈ e C

,

deg(f

◦ φ ◦ f

) = ](f

◦ φ ◦ f

)

(Q).

Since the restriction of φ◦f

to f

(U ) is injective, we deduce that deg f

◦ φ ◦ f

= 1. Thus, f

◦ φ ◦ f

is birational and so is φ.

Lemma 3.8. Let C : F (X, Y ) = 0 be a plane algebraic curve defined over K of degree N. Then there is a plane model G(X, Y ) = 0 of C defined over K with deg G = deg

G = N and

H(G) < N

H(F ), having N simple points at infinity.

P r o o f. Suppose that deg

F < N and deg

F < N . Then F (X, Y ) = X

Y

G(X, Y ) + F

(X, Y ) + . . . + F

(X, Y ), where a, b are positive integers,

G(X, Y ) = c(X + %

Y ) . . . (X + %

Y )

with c ∈ K, %

∈ K − {0} and F

(X, Y ) is a homogeneous polynomial of degree i (i = 0, . . . , N − 1). Putting X = U + mV and Y = V , where m is a non-zero integer with |m| < N/2 and G(m, 1) 6= 0, we have

F

(U, V ) = (U + mV )

V

G(U + mV, V )

+ F

(U + mV, V ) + . . . + F

(U + mV, V ), with deg

F

= N . The height of F

(U, V ) satisfies

H(F

) < (N/2)

(N − 1)!N H(F ).

Suppose next that the curve F

(U, V ) = 0 does not have N points at infinity. Write

F

(U, V ) = f

(U, V ) + . . . + f

(U, V ),

where f

(U, V ) is a homogeneous polynomial of degree i (i = 0, . . . , N ).

Putting U = 1/W , we see that the curve F

(U, V ) = 0 is birationally equiv- alent to

F

(W, V ) = f

(1, V ) + W f

(1, V ) + . . . + f

(U, V )W

= 0.

Let α be an integer with |α| < N

such that there is no ramification above W = α. Set W = T + α. It follows that the curve

F

(Π, Ξ, T ) = f

(Π, Ξ)

+ (T + αΠ)f

(Π, Ξ) + . . . + f

(Π, Ξ)(T + αΠ)

= 0 has N points with T = 0. The height of F

(Π, Ξ, T ) satisfies

H(F

) < N

N !(N + 1)H(F

) < N

H(F ).

3.3. K-rational sets. Let F (X, Y, Z) be a homogeneous absolutely irre- ducible polynomial in K[X, Y, Z] of degree N ≥ 3 such that the curve C defined by F (X, Y, Z) = 0 is of genus 0. We denote by S the set of singular points of C and for every P ∈ S let m

be the multiplicity of C at P . Suppose that C has no singularities other than ordinary multiple points. By Noether’s formula [4, Chap. 8, p. 199; 2, Chap. III, p. 614], we have

X

P ∈S

m

(m

− 1) = (N − 1)(N − 2).

Let K be an algebraic closure of K. We denote by G the Galois group of K over K. A subset E of the projective plane P

over K is called K-rational if σ(E) = E for every σ ∈ G. The set S of singular points of C is determined by equations defined over K, whence S is K-rational.

Let ν ∈ {N − 1, N − 2} and E

be a K-rational subset of C − S having

|E

| = ε

with 0 ≤ ε

≤ N − 2 and 0 ≤ ε

≤ 2N − 2. We denote by W (ν, E

) the space of homogeneous polynomials ψ(X, Y, Z) in K[X, Y, Z] of degree ν such that the curve ψ(X, Y, Z) = 0 contains every point P ∈ S with multiplicity ≥ m

− 1 and passes through the points of E

. Put δ(ν, E

) = dim W (ν, E

) and M (ν, E

) = max{H(Q)/Q ∈ S ∪ E

}. If E

= ∅, then we write W (ν) = W (ν, ∅), δ(ν) = δ(ν, ∅) and M (ν) = M (ν, ∅). We call, as usual, the points (x : y : z) on C with z = 0, points at infinity. We denote by C

the set of those points.

Lemma 3.9. Under the above assumptions, we have δ(ν, E

) = N ν − (N − 1)(N − 2) − ε

+ 1

and there is a basis {ψ

(X, Y, Z), . . . , ψ

(X, Y, Z)} of W (ν, E

), satis- fying

H(ψ

) < N

M (ν, E

)

(i = 1, . . . , δ(ν, E

)).

P r o o f. We can suppose, without loss of generality, that F (0, 1, 0) 6= 0

and that none of the points of S ∪ E

is at infinity (if this is not the case,

then we choose an appropriate projective coordinate system). Let E(X, Y, Z)

be a polynomial in W (ν). Denote by R(X) the resultant of E(X, Y, 1) and

F (X, Y, 1) with respect to Y . By [22, Theorem 5.3, p. 111], the multiplicity

of the root a of R(X) is equal to the sum of the intersection numbers of

the curves C and E(X, Y, Z) = 0 on the line X = a. Let P (i) = (α

: β

: 1) (i = 1, . . . , s) be the points of S. The polynomial Π(X) = R(X)/π(X), where

π(X) = Y

(X − α

)

,

is of degree N ν − (N − 1)(N − 2). By [4, Chap. 5, Sect. 2, p. 110] the dimension of the space W (ν) is

≥ (ν + 1)(ν + 2)

2 − (N − 1)(N − 2)

2 ≥ N − 1.

Thus, we can choose E(X, Y, Z) such that α

(i = 1, . . . , s) are not zeros of Π(X) and the discriminant of Π(X) has all its roots simple. Hence, Π(X) has N ν − (N − 1)(N − 2) zeros pairwise distinct and different from α

(i = 1, . . . , s), whence the curve E(X, Y, Z) = 0 intersects C in N ν − (N − 1)(N − 2) pairwise distinct points apart from the points of S. Hence, W (ν) cuts out on C a linear series of order N ν −(N −1)(N −2) and no cycles of this series contain points of S. By [19, Chap. XII, Sect. 4, p. 379], this linear series is complete. Since C is of genus 0, [22, Chap. VI, Sect. 7, p. 187]

implies that its dimension is N ν − (N − 1)(N − 2). It follows that W (ν, E

) cuts out on C a linear series g

of order n = N ν − (N − 1)(N − 2) − ε

and dimension r ≥ N ν − (N − 1)(N − 2) − ε

. By [22, Chap. VI, Theorem 2.5, p. 168], we have r = n = N ν − (N − 1)(N − 2) − ε

. Therefore δ(ν, E

) = N ν − (N − 1)(N − 2) − ε

+ 1.

For every P ∈ S ∪ E

we write P = (x

: y

: z

) with one of x

, y

, z

being equal to 1. By [22, Theorem 2.4, p. 55], W (ν, E

) is the space of polynomials

G(X, Y, Z) = X

a

X

Y

Z

with coefficients in K such that

G(x

, y

, z

) = 0 for every Q ∈ E

and

G

(x

, y

, z

) = 0

for every P ∈ S and every triple of non-negative integers α, β, γ with α+β + γ = m

− 2. Thus, we have a linear system in unknowns a

. The number of unknowns is (ν + 1)(ν + 2)/2. It follows that the rank % of the matrix of the above system is

% = (N − 1)(N − 2)

2 + ε

.

We consider % rows of this matrix, A

, . . . , A

, which are linearly indepen-

dent. Since S ∪E

is K-rational, Lemma 3.6 implies that there exists a basis

{ψ

(X, Y, Z), . . . , ψ

(X, Y, Z)} of W (ν, E

) satisfying H(ψ

) < %!H(A

) . . . H(A

) (i = 1, . . . , δ(ν, E

)).

We easily deduce

H(A

) < ν!M (ν, E

)

. Thus

H(ψ

) < %!ν!M (ν, E

)

(i = 1, . . . , δ(ν, E

)).

For every λ = (λ

, . . . , λ

) ∈ K

we set

φ

(X, Y, Z) = λ

ψ

(X, Y, Z) + . . . + λ

ψ

(X, Y, Z) and we denote by κ(λ) the curve defined by the equation φ

(X, Y, Z) = 0. If C

and C

are two curves in P

, we denote by I(P, C

∩C

) their intersection number at the point P of P

. For every positive integer r, we define B(r) to be the set

B(r) = {(x

, . . . , x

) ∈ Z

| |x

| ≤ r, j = 1, . . . , δ(ν, E

)}.

Lemma 3.10. Let Γ (r) be the set of δ(ν, E

)-tuples λ ∈ B(r) such that the curve κ(λ) fails at least one of the following properties:

(a) I(P, C ∩ κ(λ)) = m

(m

− 1) for every P ∈ S.

(b) I(P, C ∩ κ(λ)) = 1 for every P ∈ E

.

(c) I(P, C ∩ κ(λ)) = 0 for every P ∈ C

− (S ∪ E

).

(d) The point (0 : 1 : 0) is not on κ(λ).

Then the number of elements of Γ (r) is

≤ (2r + 1)



|S| + N + 1 + 2ε

+ X

P ∈S

m

 .

P r o o f. Set n

= m

− 1 for every P ∈ S and n

= 1 for every P ∈ E

. Suppose that there is Q ∈ S ∪ E

such that for every k ∈ {1, . . . , δ(ν, E

)}

the curve ψ

(X, Y, Z) = 0 has multiplicity > n

at Q. Consider δ(ν, E

) − 1 arbitrary points Q

, . . . , Q

on C −(S ∪E

) (j = 1, . . . , δ(ν, E

)−1).

Then there is µ ∈ K

such that the curve κ(µ) passes through the points of S ∪ E

and Q

, . . . , Q

. By Bezout’s theorem,

X

R

I(R, C ∩ κ(µ)) = N ν.

On the other hand, since the multiplicity of κ(µ) at Q is > n

, we have X

R

I(R, C ∩κ(µ)) > X

P ∈S

m

(m

−1)+ε

+N ν −(N −1)(N −2)−ε

= N ν, which is a contradiction. So, for every P ∈ S ∪ E

there is j(P ) ∈ {1, . . . . . . , δ(ν, E

)} such that the curve ψ

(X, Y, Z) = 0 has multiplicity n

at P .

Let P ∈ S ∪ E

with P = (x

: y

: 1). For every j ∈ {1, . . . , δ(ν, E

)}

and k ∈ {0, . . . , n

} we put

ψ(P, j, k) = ψ

(x

, y

, 1).

Then there is j(P ) ∈ {1, . . . , δ(ν, E

)} and k(P ) ∈ {0, . . . , n

+ 1} such that ψ(P, j(P ), k(P )) 6= 0.

If P = (x

: 1 : 0) or (1 : 0 : 0), then we define the quantity ψ(P, j, k) to be ψ

(x

, 1, 0) or ψ

(1, 0, 0) respectively. For every δ(ν, E

)-tuple λ = (λ

, . . . , λ

) in K

we set

Λ

(λ) = λ

ψ(P, 1, k(P )) + . . . + λ

ψ(P, δ(ν, E

), k(P )).

The number of solutions λ ∈ B(r) of the equation Λ

(λ) = 0 is ≤ (2r + 1)

. Note that if Λ

(λ) 6= 0, then the multiplicity of k(λ) at P is n

.

If f (X, Y ) ∈ K[X, Y ] and Q is a point on the curve f (X, Y ) = 0 which is not at infinity, then we write

T

(f (X, Y ), Q)(λ, µ) = X

s!

(s − i)!i! f

(Q)λ

µ

. Let P ∈ S − C

. Since P is an ordinary multiple point, we have

T

(F (X, Y, 1), P )(λ, µ) = (α

λ + β

µ) . . . (α

λ + β

µ),

where the factors α

λ + β

µ (i = 1, . . . , m

) are pairwise distinct. Further- more,

T

(φ

(X, Y, 1), P )(−β

, α

) = X

k

λ

T

(ψ

(X, Y, 1), P )(−β

, α

) (j = 1, . . . , m

).

For λ = (λ

, . . . , λ

) ∈ K

we write L

(λ) = X

k

λ

T

(ψ

(X, Y, 1), P )(−β

, α

).

Hence, the curves C and κ(λ) have distinct tangents at P if and only if L

(λ) 6= 0 (j = 1, . . . , m

).

If P is a point of S at infinity, then we consider the polynomial F (X, 1, Z) or F (1, Y, Z). Let now P ∈ E

− C

. Then P is a non-singular point of C and thus

T

(F (X, Y, 1), P )(λ, µ) = ζλ + ηµ.

Set

L

(λ) = X

k

λ

T

(ψ

(X, Y, 1), P )(−ζ, η).

The curves C and κ(λ) have distinct tangents at P if and only if L

(λ) 6= 0.

Hence, for P ∈ E

we have the linear equation L

(λ) = 0 and for every P ∈ S the m

linear equations

L

(λ) = 0 (j = 1, . . . , m

),

in unknowns λ

, . . . , λ

. The number of solutions λ ∈ B(r) of each of the above equations is ≤ (2r+1)

. Note that if P ∈ S with L

(λ) 6=

0 (j = 1, . . . , m

) and Λ

(λ) 6= 0, then I(P, C ∩ κ(λ)) = m

(m

− 1).

Similarly, if P ∈ E

with L

(λ) 6= 0 and Λ

(λ) 6= 0, we get I(P, C ∩ κ(λ)) = 1.

Let F

(X, Y ) be the homogeneous part of degree N of F (X, Y, 1). The points at infinity of C are S

= (a

: b

: 0) with F

(a

, b

) = 0 (i = 1, . . . , s).

Let ψ

(X, Y ) be the homogeneous part of degree ν of ψ

(X, Y, 1). For λ = (λ

, . . . , λ

) in K

we write

Θ

(λ) = λ

ψ

(a

, b

) + . . . + λ

ψ

(a

, b

) (i = 1, . . . , s).

Then φ

(S

) = 0 if and only if Θ

(λ) = 0. The number of solutions λ ∈ B(r) of the equation Θ

(λ) = 0 is ≤ (2r + 1)

. Finally, φ

(0, 1, 0) 6= 0 if and only if

λ

ψ

(0, 1) + . . . + λ

ψ

(0, 1) 6= 0.

Combining the above estimates yields the lemma.

Proposition 3.1. Let Σ be a finite subset of C. Then there is λ ∈ B(r), where

r = 1 2

 X

P ∈S

m

+ |Σ| + N

+ |S| + 2N + 2ε

 + 1,

such that the curve κ(λ) meets C in δ(ν, E

) − 1 distinct points Q

, . . . . . . , Q

which are not in S ∪ E

∪ Σ ∪ C

and satisfy

H(Q

) < ΞH(F )

M (ν, E

)

, where

Ξ ≤ N

(δ(ν, E

)r)

. P r o o f. Let r be a positive integer with

r > 1 2



|S| + N + 2ε

+ X

P ∈S

m

 .

Then the set Γ (r) of Lemma 3.10 is a proper subset of B(r). Hence, there

exists λ = (λ

, . . . , λ

) ∈ B(r) such that the curve κ(λ) : φ

(X, Y, Z) =

0 has the properties (a), (b), (c) and (d) of Lemma 3.10. Let T be the set of

those points of intersection of C and κ(λ) which are not contained in S ∪E

. By (c), the points of T are not at infinity. Bezout’s theorem yields

X

Q∈T

I(Q, C ∩ κ(λ)) = N ν − X

P ∈S∪E_ν

I(P, C ∩ κ(λ))

= N ν − (N − 1)(N − 2) − ε

= δ(ν, E

) − 1.

We can suppose, without loss of generality, that F (0, 1, 0) 6= 0. Denote by R(X) the resultant of φ

(X, Y, 1) and F (X, Y, 1) with respect to Y . By [21, Theorem 5.3, p. 111], the multiplicity of the root a of R(X) is equal to the sum of the intersection numbers of C and κ(λ) on the line X = a. Let P (i) = (a

: b

: 1) (i = 1, . . . , s) be the points of S − C

and P (i) = (a

: b

: 1) (i = s + 1, . . . , t) be the points of E

− C

. Put

π(X) = Y

(X − a

)

Y

(X − a

) and consider the polynomial

Π(X) = R(X) π(X)

= s

(λ

, . . . , λ

)X

+ . . . + s

(λ

, . . . , λ

).

The coefficients s

(λ

, . . . , λ

) are polynomials in λ

, . . . , λ

of degree ≤ N . The discriminant ∆

(λ

, . . . , λ

) of Π(X) is a polynomial in λ

, . . . , λ

of degree ≤ N

. We have |T | = δ(ν, E

) − 1 if and only if Π(X) has δ(ν, E

) − 1 pairwise distinct roots. Hence, |T | = δ(ν, E

)−1 if and only if s

(λ

, . . . , λ

) 6= 0 and ∆

(λ

, . . . , λ

) 6=

0. Furthermore, the number of δ(ν, E

)-tuples (λ

, . . . , λ

) ∈ B(r) such that

∆

(λ

, . . . , λ

) = 0 or s

(λ

, . . . , λ

) = 0 is at most (2r + 1)

(N

+ N ).

Denote by S

, . . . , S

the elements of Σ. The number of solutions λ ∈ B(r) of the equation φ

(S

) = 0 is ≤ (2r + 1)

. Thus, the number of δ(ν, E

)-tuples λ ∈ B(r) which do not have the required properties is

≤ (2r + 1)

Ω where

Ω =



|Σ| + N

+ |S| + 2N + 1 + 2ε

+ X

P ∈S

m

 .

Thus, if we take r = (Ω + 1)/2, then there exists λ ∈ B(r) such that the

curve φ

(X, Y, Z) = 0 intersects C in δ(ν, E

) − 1 pairwise distinct points

Q

= (x

: y

: 1) (i = 1, . . . , δ(ν, E

) − 1) which are not in S ∪ E

∪ Σ.

We may assume, without loss of generality, that one of the coefficients of each ψ

is 1. Then

H(φ

) < Ω + 1

2 δ(ν, E

)H(ψ

) . . . H(ψ

) and Lemma 3.9 yields

H(φ

) < Ω + 1

2 δ(ν, E

)(N

M (ν, E

)

)

. The resultant R(X) of F (X, Y, 1) and φ

(X, Y, 1) satisfies

R(x

) = 0 (i = 1, . . . , δ(ν, E

) − 1).

Thus, Lemma 3.1 implies

H(x

) < 2H(R) (i = 1, . . . , δ(ν, E

) − 1).

By Lemma 3.2, we have

H(R) < (N + ν)!(N + 1)

(ν + 1)

H(F )

H(φ

)

. Thus,

H(x

) < N

H(F )

H(φ

)

.

Interchanging the roles of x

and y

we obtain the same bound for H(y

).

Therefore

H(Q

) < ΞH(F )

M (ν, E

)

, where

Ξ ≤ N



δ(ν, E

) Ω + 1 2



.

Corollary 3.1. For every positive integer % there are % K-rational sub- sets Σ

(i = 1, . . . , %) of C such that Σ

∩ (S ∪ E

∪ C

) = ∅, Σ

∩ Σ

= ∅ for i 6= j, |Σ

| = δ(ν, E

) − 1, and for every Q ∈ Σ

we have

H(Q) < Ξ

H(F )

M (ν, E

)

, where

Ξ

≤ N

δ(ν, E

)

4

×



N

+ |S| + 2N + 2ε

+ X

P ∈S

m

+ (i − 1)(δ(ν, E

) − 1) + 2



.

P r o o f. For Σ = ∅, Proposition 3.1 implies that there is λ ∈ B(r

), where

r

= 1 2



N

+ |S| + 2N + 2ε

+ X

P ∈S

m



+ 1,

such that the curve φ

(X, Y, Z) = 0 meets C in δ(ν, E

)−1 pairwise distinct points Q

, . . . , Q

which are not in S ∪ E ∪ C

and satisfy

H(Q

) < Ξ

H(F )

M (ν, E

)

, where

Ξ

≤ N

(δ(ν, E

)r

)

.

Since F (X, Y, Z) and φ

(X, Y, Z) are in K[X, Y, Z], the intersection of the two curves is K-rational. In addition, S ∪ E

is K-rational. Hence, so is {Q

, . . . , Q

}.

Next, take Σ

= {Q

, . . . , Q

}. Proposition 3.1 implies that there exists µ ∈ B(r

), where

r

= 1 2



δ(ν, E

) − 1 + N

+ |S| + 2N + 2ε

+ X

P ∈S

m

 + 1, such that the curve φ

(X, Y, Z) = 0 meets C in δ(ν, E

)−1 pairwise distinct points S

, . . . , S

which are not at infinity and satisfy

H(S

) < Ξ

H(F )

M (ν, E

)

, where

Ξ

≤ N

(δ(ν, E

)r

)

.

Furthermore, the points S

, . . . , S

are not in S ∪ E

∪ Σ

. Since S, E

and Σ

are K-rational, so is {S

, . . . , S

}. Repeating the above procedure yields the assertion.

3.4. Proof of Theorem 3.1. Take ν = N − 2 and E

= ∅. Corollary 3.1 implies that there are two K-rational subsets Σ

(i = 1, 2) of C with |Σ

| = N − 2 and Σ

∩ Σ

= ∅ such that for every Q ∈ Σ

∪ Σ

we have

H(Q) < N

H(F )

M (N − 2)

. By Lemma 3.4,

M (N − 2) < 4(N + 1)

H(F )

. Thus, for every Q ∈ Σ

∪ Σ

we have

H(Q) < (N + 1)

H(F )

.

Next, let ν = N − 1 and E

= Σ

∪ Σ

. Then Lemma 3.7 implies that there is a basis {ψ

(X, Y, Z), ψ

(X, Y, Z), ψ

(X, Y, Z)} of W (N −1, Σ

∪Σ

) such that

H(ψ

) < N

M (N − 1, E

)

(i = 1, 2, 3).

Combining Lemma 3.4 and the above inequalities, we get

H(ψ

) < (N + 1)

H(F )

(i = 1, 2, 3).

The set of common zeros of ψ

(X, Y, Z), ψ

(X, Y, Z), ψ

(X, Y, Z) in P

is V = Σ

∪ Σ

∪ S. Put U = C − V and consider the morphism ψ : U → P

given by

ψ(P ) = (ψ

(P ) : ψ

(P ) : ψ

(P )) for every P ∈ U.

We denote by e C the closure of ψ(U ) in P

. The morphism ψ defines a rational map Ψ from C to e C. First, we prove that e C is a conic.

Consider the set e S of singular points of ψ(U ). By Proposition 3.1, there is λ = (λ

, λ

, λ

) ∈ Z

such that the curve κ(λ) defined by

φ

(X, Y, Z) = λ

ψ

(X, Y, Z) + λ

ψ

(X, Y, Z) + λ

ψ

(X, Y, Z) = 0 meets C in two distinct points Γ

, Γ

which are not in S ∪ E

∪ ψ

( e S).

Thus, the points ψ(Γ

) and ψ(Γ

) are simple and I(Γ

, C ∩ κ(λ)) = 1 (i = 1, 2). Denote by L the line in P

defined by

ε(X, Y, Z) = λ

X + λ

Y + λ

Z = 0.

Put ∆

= ψ(Γ

) (i = 1, 2). Then L ∩ e S = {∆

, ∆

}. Let O

( e C) be the local ring of e C at ∆

and O

(C) be the local ring of C at Γ

(i = 1, 2). The morphism ψ induces a ring monomorphism ψ∗ : O

( e C) → O

(C) given by

ψ ∗ (f ) = f ◦ ψ for every f ∈ O

( e C).

Since Γ

and ∆

are non-singular points of C and e C respectively, it follows that O

( e C) and O

(C) are discrete valuation rings. We denote by ord

and ord

the order functions defined by O

(C) and O

( e C) respectively.

Then

ord

(ψ ∗ (ε)) = ord

(φ

) = I(Γ

, C ∩ κ(λ)) = 1.

Since ord

(ε) ≤ ord

(ψ ∗ (ε)), it follows that ord

(ε) = 1, whence I(∆

, e C ∩ L) = 1 (i = 1, 2). By Bezout’s theorem, we get

deg e C = I(∆

, e C ∩ L) + I(∆

, e C ∩ L) = 2.

Hence e C is a conic.

Let G(X, Y, Z) = 0 be the equation defining e C. We now calculate an upper bound for H(G). Let (x, y) ∈ K

satisfy F (x, y, 1) = 0 and ψ

(x, y, 1) 6= 0. Put

µ = ψ

(x, y, 1)

ψ

(x, y, 1) , ξ = ψ

(x, y, 1) ψ

(x, y, 1) . Then (x, y, µ) is a solution of the system

F (X, Y, 1) = 0, Ψ

(X, Y, M ) = ψ

(X, Y, 1) − M ψ

(X, Y, 1) = 0 and (x, y, ξ) is a solution of the system

F (X, Y, 1) = 0, Ψ

(X, Y, Ξ) = ψ

(X, Y, 1) − Ξψ

(X, Y, 1) = 0.